Hello Rebecca,
I will try to give you an explanation for how to multiply polynomials.
(x-4)^2 (x-9)
= (x-4)(x-4)(x-9)
You can do the multiplication in any order. Just for fun let's do (x-4)(x-9) first. What you should remember is that multiplication for polynomials is distributive.
The distributive law says that a(b+c) = a*b+a*c. You can also apply it to (a+b)(c+d) by distributing twice, first a then b: (a+b)(c+d) = a(c+d) +b(c+d) = ac+ad + bc + bd.
What this means is that (x-4)(x-9) = x(x-9) - 4(x-9) notice that I get x and -4 from (x-4).
Now x(x-9) also requires the distributive property: you do: x(x-9) = x*x - x*9 = x^2-9x
Likewise for 4(x-9) you get: 4(x-9) = 4*x - 4*9=4x-36. Now recall that these are meant to go together:
x^2-9x - [4x-36] = x^2-9x -4x + 36 = x^2-13x+36
Now we need to multiply (x-4)(x^2 - 13x +36).
Again break it up into parts: x(x^2-13x+36) - 4(x^2 - 13x +36)
And again first distribute the x: x(x^2 - 13x +36) = x^3-13x^2+36x
Then distribute the 4: 4(x^2 - 13x +36) = 4x^2 -52x +144
And finally put them together:
x^3-13x^2+36x - [4x^2 - 52x +144]
= x^3-13x^2+36x -4x^2+52x - 144
= x^3 -17x^2 +88x -144
The FOIL method is a mnemonic for remembering that this is what you are doing when you multiply polynomials:
(a+b)(c+d) =
a(c+d) + b(c+d) =
ac + ad + bc + bd
First Outside Inside Last
I hope this helps in understanding the FOIL method. Remember that you could use the same strategy for numbers since they also follow the distributive law: 4(3+4) = 4*3+4*4 = 12 + 16 = 4(7)= 28, and perhaps a little harder; 12*12 = (10+2) * (10+2) = 10(10+2)
+ 2(10+2) = 100 + 20 +20 + 4 = 144.
If you have any questions, please feel free to ask!
Sincerely,
Jose
The FOIL method is a mnemonic for remembering that this is what you are doing when you multiply polynomials:(a+b)